variations of the turing machine - florida tech department ...ryan/cse4083/busch/class17.pdf ·...
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Variationsof the
Turing Machine
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Read-Write Head
Control Unit
a a c b a cb b a a
Deterministic
The Standard Model
Infinite Tape
(Left or Right)
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Variations of the Standard Model
• Stay-Option
• Semi-Infinite Tape
• Off-Line
• Multitape
• Multidimensional
• Nondeterministic
Turing machines with:
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We want to prove:
Each Class has the same
power with the Standard Model
The variations form different
Turing Machine Classes
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Same Power of two classes means:
Both classes of Turing machines accept
the same languages
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Same Power of two classes means:
For any machine of first class 1M
there is a machine of second class 2M
such that: )()( 21 MLML
And vice-versa
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a technique to prove same powerSimulation:
Simulate the machine of one class
with a machine of the other class
First Class
Original Machine
1M 1M
2M
Second Class
Simulation Machine
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Configurations in the Original Machine
correspond to configurations
in the Simulation Machine
nddd 10Original Machine:
Simulation Machine: nddd 10
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The Simulation Machine
and the Original Machine
accept the same language
fdOriginal Machine:
Simulation Machine:fd
Final Configuration
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Turing Machines with Stay-Option
The head can stay in the same position
a a c b a cb b a a
Left, Right, Stay
L,R,S: moves
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Example:
a a c b a cb b a a
Time 1
b a c b a cb b a a
Time 2
1q 2q
1q
2q
Sba ,
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Stay-Option Machines
have the same power with
Standard Turing machines
Theorem:
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Proof:
Part 1: Stay-Option Machines
are at least as powerful as
Standard machines
Proof: a Standard machine is also
a Stay-Option machine
(that never uses the S move)
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Part 2: Standard Machines
are at least as powerful as
Stay-Option machines
Proof: a standard machine can simulate
a Stay-Option machine
Proof:
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1q 2qLba ,
1q 2qLba ,
Stay-Option Machine
Simulation in Standard Machine
Similar for Right moves
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1q 2qSba ,
1q 2qLba ,
3qRxx ,
Stay-Option Machine
Simulation in Standard Machine
For every symbol x
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Example
a a b a
1q
Stay-Option Machine:
1
b a b a
2q
2
1q 2qSba ,
Simulation in Standard Machine:
a a b a
1q
1
b a b a
2q
2
b a b a
3q
3
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Standard Machine--Multiple Track Tape
bd
a
bba
ac
track 1
track 2
one symbol
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bd
a
bba
ac
track 1
track 2
1q 2qLdcab ),,(),(
1q
bd
a
b
c
d
ac
track 1
track 2
2q
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Semi-Infinite Tape
.........# a b a c
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Standard Turing machines simulate
Semi-infinite tape machines:
Trivial
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Semi-infinite tape machines simulate
Standard Turing machines:
Standard machine
.........
Semi-infinite tape machine
..................
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Standard machine
.........
Semi-infinite tape machine with two tracks
..................
reference point
#
#
Right part
Left part
a b c d e
ac b
d e
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1q2q
Rq2Lq1
Lq2 Rq1
Left part Right part
Standard machine
Semi-infinite tape machine
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1q 2qRga ,
Standard machine
Lq1Lq2
Lgxax ),,(),(
Rq1Rq2
Rxgxa ),,(),(
Semi-infinite tape machine
Left part
Right part
For all symbols x
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Standard machine.................. a b c d e
1q
.........
Semi-infinite tape machine
#
#
Right part
Left part ac b
d e
Lq1
Time 1
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Time 2
g b c d e
2q
#
#
Right part
Left part gc b
d e
Lq2
Standard machine..................
.........
Semi-infinite tape machine
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Lq1Rq1
R),#,(#)#,(#
Semi-infinite tape machine
Left part
At the border:
Rq1Lq1
R),#,(#)#,(# Right part
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.........
Semi-infinite tape machine
#
#
Right part
Left part gc b
d e
Lq1
.........#
#
Right part
Left part gc b
d e
Rq1
Time 1
Time 2
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Theorem: Semi-infinite tape machines
have the same power with
Standard Turing machines
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The Off-Line Machine
Control Unit
Input File
Tape
read-only
a b c
d eg
read-write
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Off-line machines simulate
Standard Turing Machines:
Off-line machine:
1. Copy input file to tape
2. Continue computation as in
Standard Turing machine
331. Copy input file to tape
Input Filea b c
Tape
a b c
Standard machine
Off-line machine
a b c
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2. Do computations as in Turing machine
Input Filea b c
Tape
a b c
a b c
1q
1q
Standard machine
Off-line machine
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Standard Turing machines simulate
Off-line machines:
Use a Standard machine with four track tape
to keep track of
the Off-line input file and tape contents
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Input Filea b c
Tape
Off-line Machine
e f gd
Four track tape -- Standard Machine
a b c d
e f g0 0 0
0 0
1
1
Input File
head position
Tapehead position
##
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a b c d
e f g0 0 0
0 0
1
1
Input File
head position
Tapehead position
##
Repeat for each state transition:• Return to reference point
• Find current input file symbol
• Find current tape symbol
• Make transition
Reference point
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Off-line machines
have the same power with
Stansard machines
Theorem:
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Multitape Turing Machines
a b c e f g
Control unit
Tape 1 Tape 2
Input
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a b c e f g
1q 1q
a g c e d g
2q 2q
Time 1
Time 2
RLdgfb ,),,(),( 1q 2q
Tape 1 Tape 2
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Multitape machines simulate
Standard Machines:
Use just one tape
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Standard machines simulate
Multitape machines:
• Use a multi-track tape
• A tape of the Multiple tape machine
corresponds to a pair of tracks
Standard machine:
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a b c h e f g
Multitape MachineTape 1 Tape 2
Standard machine with four track tape
a b c
e f g0 0
0 0
1
1
Tape 1
head position
Tape 2head position
h0
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Repeat for each state transition:
•Return to reference point
•Find current symbol in Tape 1
•Find current symbol in Tape 2
•Make transition
a b c
e f g0 0
0 0
1
1
Tape 1
head position
Tape 2head position
h0
####
Reference point
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Theorem: Multi-tape machines
have the same power with
Standard Turing Machines
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Same power doesn’t imply same speed:
Language }{ nnbaL
Acceptance Time
Standard machine
Two-tape machine
2n
n
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}{ nnbaL
Standard machine:
Go back and forth times 2n
Two-tape machine:
Copy to tape 2 nb
Leave on tape 1 na
Compare tape 1 and tape 2
n( steps)
n( steps)
n( steps)
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MultiDimensional Turing Machines
x
y
a
b
c
Two-dimensional tape
HEADPosition: +2, -1
MOVES: L,R,U,D
U: up D: down
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Multidimensional machines simulate
Standard machines:
Use one dimension
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Standard machines simulate
Multidimensional machines:
Standard machine:
• Use a two track tape
• Store symbols in track 1
• Store coordinates in track 2
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x
y
a
b
c
a
1b
#symbols
coordinates
Two-dimensional machine
Standard Machine
1 # 2 # 1
c
# 1
1q
1q
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Repeat for each transition
• Update current symbol
• Compute coordinates of next position
• Go to new position
Standard machine:
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MultiDimensional Machines
have the same power
with Standard Turing Machines
Theorem:
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NonDeterministic Turing Machines
Lba ,
Rca ,
1q
2q
3q
Non Deterministic Choice
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a b c
1q
Lba ,
Rca ,
1q
2q
3q
Time 0
Time 1
b b c
2q
c b c
3q
Choice 1 Choice 2
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Input string is accepted if
this a possible computation
w
yqxwq f
0
Initial configuration Final Configuration
Final state
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NonDeterministic Machines simulate
Standard (deterministic) Machines:
Every deterministic machine
is also a nondeterministic machine
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Deterministic machines simulate
NonDeterministic machines:
Keeps track of all possible computations
Deterministic machine:
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Non-Deterministic Choices
Computation 1
1q
2q
4q
3q
5q
6q 7q
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Non-Deterministic Choices
Computation 2
1q
2q
4q
3q
5q
6q 7q
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• Keeps track of all possible computations
Deterministic machine:
Simulation
• Stores computations in a
two-dimensional tape
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a b c
1q
Lba ,
Rca ,
1q
2q
3q
Time 0
NonDeterministic machine
Deterministic machine
a b c
1q
# # # # ##### # #
##
# #
Computation 1
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Lba ,
Rca ,
1q
2q
3q
b b c2q
# # # # #### #
#
# #
Computation 1
b b c
2q
Choice 1
c b c
3q
Choice 2
c b c3q ##
Computation 2
NonDeterministic machine
Deterministic machine
Time 1
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Repeat
• Execute a step in each computation:
• If there are two or more choices
in current computation:
1. Replicate configuration
2. Change the state in the replica
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Theorem: NonDeterministic Machines
have the same power with
Deterministic machines
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Remark:
The simulation in the Deterministic machine
takes time exponential time compared
to the NonDeterministic machine
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Polynomial Time in NonDeterministic Machine:
NP-Time
Polynomial Time in Deterministic Machine:
P-Time
Fundamental Problem: P = NP ?